Generic representations of the finite general linear groups and the Steenrod algebra. III (Q1897329)

[For part II cf. ibid. 8, No. 4, 395-428 (1994; Zbl 0830.20065).] Let \({\mathcal F} (q)\) be the abelian category with objects the functors \(F:\) finite \({\mathbf F}_q\)-vector spaces \(\to{\mathbf F}_q\)-vector spaces and with morphisms the natural transformations. The groups \(\text{Ext}^s_{{\mathcal F}(q)}(F,G)\) are conjecturally \(\text{Ext}^s_{\text{GL}(V)}(F(V),G(V))\) for finite \(F\) and \(G\) and large \(V\). The author develops some tools for studying such groups. In particular, with \(S^n\) defined by \(S^n(V)=V^{\otimes n}/\Sigma_n\), and with \(\Phi:S^n\to S^{qn}\) the \(q\)th power map, the following ``vanishing theorem'' is proved: for all finite \(F\in{\mathcal F}(q)\) and \(s>0\), one has \[ \text{colim}\{\text{Ext}^s_{{\mathcal F}(q)}(F,S^n)@>\Phi>>\text{Ext}^s_{{\mathcal F}(q)}(F,S^{qn})@>\Phi>>\dots\}=0. \] As an application, the space of natural transformations from the \(m\)th symmetric invariant functor to the \(n\)th symmetric coinvariant functor is calculated for all \(m\) and \(n\).